Microwave range hood install -- stud in middle of...

You have a few options, First off, I am assuming that the stud you refer to is an outside wall. Assuming that is true, then you have a few options.... one of them is to cut thru the stud and run the vent out. Even if that stud is on a bearing wall ( meaning it is carrying a roof load above it) ... It is unlikely to pose a problem if that one stud has been compromised with it cut out, If that concerns you.. and you are handy, you can open that area of wall up from the inside and box the area that the vent will be located in with short pieces of 2x4 to the studs on either side of it... Another option is to vent the hood up thru the cabinet above and direct that out slightly offset to the stud in the wall..
One thing I have done is to cut the stud out and then to place a piece of 1/2" plywood in the area that the micro/hood will be located and attach that piece with drywall screws to the studs in the wall.. That piece of plywood will help support the stud you cut when you screw it to the adjacent studs and also act as a place to secure the hood mounting bracket on the wall. Granted, it shifts the hood out from the wall 1/2 " but then you have plenty of surface to secure the plate and don't have to worry about hitting a stud or using wall anchors.. I hope you understand my comments..If not...please respond to this and I will provide additional information to clarify this... Hope this helps !