https://www.google.com/maps/dir/George+Bush+Intercontinental+Airport,+2800+N+Terminal+Rd,+Houston,+TX+77032/Webster,+TX/@29.7591552,-95.5259602,10z/data=!3m1!4b1!4m14!4m13!1m5!1m1!1s0x8640b423d94355c7:0x9cc6b4fcc8c0c231!2m2!1d-95.336783!2d29.99022!1m5!1m1!1s0x86409ce96824a435:0xbb5edb33badfa46!2m2!1d-95.1182645!2d29.5377315!3e0
Looks like no tolls, but thats a guess.
Concur, no tolls (at least for now). This is a good route.