There's a few well known tricks that will help you deal with the 'Liege Wants your Troops' event.
A: If you agree to letting him have your troops, always set your Troop Maintenance Slider to 0. Your Liege will pay for those troops and they won't take a hit for loss of wages and you won't go into debt paying for HIS war.
B: If you don't want him to have those troops, but you aren't ready to deal with the 'loyalty hits' (which if it gets low enough, might give you the Rebellious trait and those events leading from that), then DON'T click yes or no. You've got about a 2 month grace period so you can now do one of two things.
1. Once the Liege asks for your troops, pause the game, mobilize them yourself, set Maintenance to 0, click YES then just leave those troops where they are. In most cases those individual garrisons can easily fit under the 'troop maintenance' for those provinces without attrition at 0 wages. If you mobilize them before clicking yes, then you retain full control over those troops (and full responsibility for their costs).
2. If for some reason you don't want those troops sitting around (province is in rebellion/disease/etc), leave that request alone, then wait until 2 or 3 days is left of that 2 month timer, pause the game, raise your troops, click YES, then immediately de-mobilize those troops. Your Liege will most likely ask for them again in the next few days, but you once again have 2 months before you need to repeat these steps.