"a quick peek" is correct and usable in written English. This phrase is typically used to describe when someone glances at something quickly. For example: "She took a quick peek at the painting before deciding whether to buy it.".
I can't resist having a quick peek.
Here's a quick peek at some of our newest features.
They take a quick peek, find it empty and walk away.
Poll workers, incoming voters or anyone else wandering by could have a quick peek.
And a quick peek into the basement reveals just how deep that allegiance runs.
We take a quick peek at the nests, then move on".
But the Labour leadership election keeps pulling us in for a quick peek.
Awesome tool! I started using it one year ago and I never had to look for another app
Ha Thuy Vy
MA of Applied Linguistic, Maquarie University, Australia