Yes, there is a link between the Spanish word pregunta and the English words pregnant and embarrassed. However, it is (perhaps) not the link you were expecting.
All three words have, as a historical meaning, the idea of hindering or impeding.
Embarrass has, at its root, the idea of impeded or hindered, which has the word bar at its root (as in bar the way). In Spanish the word came to describe a woman who is burdened with pregnancy. In English, the word describes a person burdened with a social problem.
Preguntar, to ask or inquire, comes from the Latin percontari, which meant question, inquire, investigate. Percontari comes from contor, which meant hinder, delay, impede. Apparently, the Romans considered an investigation to be something which delays progress. Did they favor results over analysis?
Pregnant comes from a word meaning pressed, related to the word oppressed (burdened). This is (and was) another way of saying hindered or impeded.
So, yes, the words tell a story and can be linked. The Romans were fond of puns, so I believe that they would approve linking the concepts of “social awkwardness,” “pregnant,” and “having trouble with mobility.”
I hope this helps.