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.