Will calamari trigger a shellfish allergy?

Yes. About 10-15% of people who are shellfish allergic (shrimp, lobster, etc) are also clinically sensitive to mollusks (clams, squid(calamari), octopus). Ask your allergist before introducing mollusks to your diet.
It can. Calamari (squid/octopus) is not typically thought of as a shellfish but it is a mollusc - in the same family as oysters and mussels.