Will a short/ground in the thermostat wires cause a model(-) HQA to burn up the 24 volt transformer

The fuse protects the transformer from being damaged due to too much amperage, this is typically due to the circuit having a short or a device in that circuit being shorted.
See below the transformer's relationship to the thermostat.
The RED wire is the "hot leg" of the 24 volt power provided by the transformer to the thermostats R or RED terminal as pictured below.
The 24 volt hot leg enters the thermostat on R or RED.
The various other terminals White = heat, Yellow = cool ,
Green = fan and Common is the side of the 24 volt power source that every 24 volt circuit terminates or return to complete the circuit.
Thus the common name designated, note hoe everything returns to Common.
Well if we jumped Red directly to Common that is a short and will blow the fuse as you observe.
See the 2 diagrams in red describing a notrmal circuit then a short circuit.

you have a 24 volt shorts check your thermostat wires to see if they are frayed or if they are touching anything. Check your outside wires also go to the thermostat.

One or more of the solenoids operated switches is your most probable cause. If you have a clamp on amp meter, measure the amps pulled by each component with a solenoid. The reversing valve solenoid and the main contactor solenoid. Each one of the these components are typically rated at 6watts. If your control voltage is around 25 volts, your amp draw should be about .24 amps max. If your transformer cannot be reset without blowing, install a 3 amp inline fuse so when the overload occurs the only thing that blow is an inexpensive fuse costing about 25 cents instead of a several dollar transformer. If your fuse/transformer immediately blows, try looking for a direct ground in the control wiring. If problems has persisted for a while, you may even have to look into replacing your thermostat wire. The control wire travels over the thermostat wire throught the thermostat and excess amps through the thermostat wire can break down the insulation jacket and short to the common wire causing the control transformer to blow. Another thing to check is the condition of your thermostat wire to check if the insulation is shredding off. Old thermostat wire, especially thermostat wire exposed to sunlight deteriorates quicker. Hope this helps.

Go to your thermostat and look at the FAN switch, it will be in the AUTO position typically.
Set the FAN switch to the ON setting,
by doing this if the fan motor operates you know that the High voltage & the low voltage (24 volt control power) are present.
This saves you from having to go to the breaker panel and checking / resetting the breaker as well as checking the Transformer and automotive type 3 amp control circuit fuse (generally on the circuit board).

If you do not get any fan operation by switching the fan switch to the ON setting, you then will have to determine if the breaker, transformer and aforementioned 3 amp fuse are all good.

You will need a volt meter to test further.
At the thermostat remove the cover thus exposing the thermostats sub base and assorted circuits/wiring.

RED or R
Red is the 24 volt "hot leg" of 24 volt power which originates at the transformer. Red enters the thermostat on the Red or R terminal, some thermostats will have an RC and an RH terminal, these are jumpered together on single transformer systems as they are for Red Cool and Red Heat, without a jumper wire on RC and RH, the Red "hot leg" of the 24 volt control voltage will only energize the terminal its wired to, some stats are battery powered and do not use the transformers 24 volt power (parasitically) to power the thermostat.
If the thermostat is powered by the transformers 24 volt power, there will be a "COMMON" wire on the common terminal of the thermostat, this is the other side of the 24 volt power from the transformer, the side opposite from RED the 24 volt "hot leg"
Common is called common as its the side of power that EVERY
24 volt circuit terminates, or completes its circuit, thus the COMMON designation.
During a Heat call the 24 volt hot leg is sent out via the white/heat terminal to the gas valve etc.
So between terminals Common & White , it should read 24 volts during a call for heat.
Same for Fan which is the Green or G terminal,
Same for Cool which is the Yellow or Y terminal.
If you have no power to these terminals when calling for heat or cool or fan then yes the thermostat is defective and requires replacement.
DO NOT let the RED wire touch ground or the COMMON terminal, this is a direct short and blows the fuse or transformer
if not equipped with fuse protection.

Normally you should read 24 volts between RED and COMMON

If it is happening only when the cooling is being turned on, it will more than likely be between the indoor unit and the outdoor unit. There is small thermostat wiring that is wired between your indoor blower or furnace and the wire runs to the outdoor unit. A lot of times this small wiring gets chewed or freyed or a nail may be driven through it. This will cause a small short in the system that will burn up your transformer.

The only other problem could be the actual contactor this wiring goes to in the outdoor unit. Sometimes, the contactor coil may be shorted and can also cause the transformer to short out. The contactor is the small automatic switch in the outdoor unit, about 3" long and has all the wiring going to it including the high voltage wiring from the house.

Hello, if the outside unit is not running start at the outside disconnect box and verify if you are getting 240 volts, if you are continue to the contactor coil, check for 24 volts at the contactor coil if there is 24 volts and the contactor does not energize then it must be replaced. If you do not have 24 volts then the low voltage ( thermostat) wiring must be checked out for a short circuit, loose wiring, or the thermostat might be bad. Also, you can check for 24 volts at tjebcontrol board between the r and c terminal if no 24 volts then you have a bad transformer.

Typical Thermostat Wiring for 4 or 5 wire Heat/Cool Thermostat (not heat pumps)
4 Wire Termostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
with one transformer for both units
Rc One Side of 24 volt Transformer
Rh Jumper Rc to Rh
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
With 2 transformers one for indoor unit
And one for outdoor unit
Rc = Red One Side of (outside) 24 volt (outside transformer)
G = Green Blower (outside transformer)
Y = Yellow Cool (outside transformer)
Rh = Blue One side of (inside) 24 volt (inside transformer)
W = White Heat (inside transformer)
COOLING ONLY Thermostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
HEATING ONLY
R or Rh One Side of 24 volt Transformer
W = White Heat

What are you trying to wire up?
Typical Thermostat Wiring for 4 or 5 wire Heat/Cool Thermostat (not heat pumps)
4 Wire Termostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
with one transformer for both units
Rc One Side of 24 volt Transformer
Rh Jumper Rc to Rh
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
With 2 transformers one for indoor unit
And one for outdoor unit
Rc = Red One Side of (outside) 24 volt (outside transformer)
G = Green Blower (outside transformer)
Y = Yellow Cool (outside transformer)
Rh = Blue One side of (inside) 24 volt (inside transformer)
W = White Heat (inside transformer)
COOLING ONLY Thermostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
HEATING ONLY
R or Rh One Side of 24 volt Transformer
W = White Heat

Typical Thermostat Wiring for 4 or 5 wire Heat/Cool Thermostat (not heat pumps)
4 Wire Termostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
with one transformer for both units
Rc One Side of 24 volt Transformer
Rh Jumper Rc to Rh
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
With 2 transformers one for indoor unit
And one for outdoor unit
Rc = Red One Side of (outside) 24 volt (outside transformer)
G = Green Blower (outside transformer)
Y = Yellow Cool (outside transformer)
Rh = Blue One side of (inside) 24 volt (inside transformer)
W = White Heat (inside transformer)
COOLING ONLY Thermostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
HEATING ONLY
R or Rh One Side of 24 volt Transformer
W = White Heat

Typical Thermostat Wiring for 4 or 5 wire Heat/Cool Thermostat (not heat pumps)
4 Wire Termostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
with one transformer for both units
Rc One Side of 24 volt Transformer
Rh Jumper Rc to Rh
G = Green Blower
Y = Yellow Cool
W = White Heat
5 Wire Thermostat
With 2 transformers one for indoor unit
And one for outdoor unit
Rc = Red One Side of (outside) 24 volt (outside transformer)
G = Green Blower (outside transformer)
Y = Yellow Cool (outside transformer)
Rh = Blue One side of (inside) 24 volt (inside transformer)
W = White Heat (inside transformer)
COOLING ONLY Thermostat
R = Red One Side of 24 volt Transformer
G = Green Blower
Y = Yellow Cool
HEATING ONLY
R or Rh One Side of 24 volt Transformer
W = White Heat