Is it possible to get eye infection if I blinked underwater in a pool?

No. Eyes can get irritated by chlorine. Also no contact lens wear in hot tubs, otherwise increased infection risk.
Unlikely. Most municipal swimming pools have sufficient chlorine to minimize risk of ocular infection. The possibility exists but is low - may be higher with contact lens use however. Best to use a pair of goggles for extended swimming. Also helps keep irritating chlorine out of the eye as well.