I only peered / glimpsed / squinted the Queen from a distance before her car drove away
The correct choice would be: I only glimpsed the Queen from a distance before her car drove away.
The words "only" and "before" tell us that the duration of the verb was short lived. Squinted doesn't work because it needs to followed by the word "at": I only squinted at the Queen. The same rule applies for peered. Peering also usually indicates that you are looking at something or one for an extended period of time.
I hope this helps!