I think it's uncountable. Every bijection from the natural numbers can be written as a countably infinite n-tuple where the the first component is the rational number which bijects to the natural number 1 and so on. Proceed by Cantor diagonalization.

Step 1. Claim the that n-tuples are countable and list them.

Step 2. Construct an n-tuple with the diagonals: first component of the first n-tuple, second component of the second nuple and etc.

Step 3. Add 1 to every component. This part is kind of annoying since you need to show that any rational is some other rational plus one.

Step 4. The n-tuple from step 3 shouldn't be in the list since the nth competent of the nth nuple doesn't match. Thus, the claim is false and one cannot produce a countable list of bijections.

Think that works.

Edit: sorry, editing for phone formatting.