Most HVAC systems are “closed”, specifically designed so that it neither draws in air from the outside or pushes inside air to the outside when it’s in operation. The “air balance” is said to be “zero” ideally.
The HVAC returns do not exhaust outside of the house, they go through a filter and are recirculated back in to the house. That filter will do nothing for the smoke generated by a GF. If you plumbed the exhaust of your GF to the returns of your HVAC the outcome will be fairly suboptimal (under emphasizing the degree of suboptimality). All this will do is distribute the smoke evenly throughout the house.
No. You want to totally ignore your HVAC system and duct the GF out the window and run it normally. The GF will pump inside air to the outside to remove the smoke, and replacement air will leak in around doors and windows and wall outlets and bathroom ceiling fans and a myriad of other places.
Now, depending on how well the air balance is within the house, running the HVAC at the same time as the GF may reduce the efficiency of the GF exhaust system. You may have noticed that the HVAC outlets and returns aren’t always the same size. This can result in some pressure differentials in different areas of the house, for example a closed room with a larger outlet than a return (or no return) would run at higher pressure than one that had a smaller outlet and a larger return (or no outlet, only return). If your GF is in this second room and the HVAC is running, the GF will have to fight that pressure differential to pump smoke out, it’ll be like the fan isn’t running as fast. The GF is not 100% smoke leak-free even in the best of conditions, and this situation will increase the leak rate to possibly unacceptable levels.