It depends entirely on context.
The normal meaning, when speaking about "recycling paper", would refer to the environmental process of "recycling", and would mean pieces of paper.:
Recycle (definition 1) :
To treat or process (used or waste materials) so as to make suitable
Paper (definition 1):
A substance made from wood pulp, rags, straw, or other fibrous
material, usually in thin sheets
If this is the intended meaning, then- no, "I recycle a paper" is not the right wording for the sentence. "I recycle a piece of paper" or "I recyle papers" is what is needed.
However, as others have already pointed out, both words have multiple meanings. From these you could create several scenarios where the sentence "I recycle a paper" is valid.