"Pair" or "couple"?

Solution 1:

The usual term for a male and female bond is a pair not couple.

Speaking casually, "I saw a pair/couple of puffins the other day" would mean you'd seen two birds, of either sex.

As a verb, birds are said to pair, meaning male and female bonding for the purpose of rearing chicks. But to couple means to engage in sexual intercourse.

Solution 2:

In my experience pair of means 2 and couple of means 2 or more.

A pair of puffins

This would be 2 puffins.

A couple of puffins

This would be 2 or a few (small number of) puffins.

What I think you want to say is a puffin couple.

Solution 3:

The key here is word order and context, not word choice. If you look at the definitions of both words at dictionary.com (pair and couple), you'll find they actually use the other word in the definition. So instead of saying pair of, or couple of puffin, which could imply two non-related, you need to say a puffin pair or a puffin couple, which does imply relationship.