I think peek can be taken out of the confusion equation most easily. It simply means ‘a secret look’, and one cannot ‘secret look one’s interest’ and claim grammatical wherewithal. However, errors arise between peak and pique.
Peak is often used wrongly in this expression, presumably because it sounds like it’s bringing one’s interest to a peak, ‘a highest or maximum point’.
The correct phrase is ‘to pique one’s interest’.
Pique, in this context, means ‘to provoke’.
If you ever confuse peek and peak, just remember that peek is like peer, or see the ‘ee’s like a pair of eyes.
I hope this post provoked you interest. Likes, shares, comments and such will be met with eternal gratitude.