You may define a statement of a theory as "true" if it holds in any model, and as "provable" if there is a logical deduction from the axioms of the theory.

Then Goedel's first incompleteness theorem tells us, that any theory, which is as least as powerful as number theory, contains true statements which are not provable.

Hence "true" does not imply "provable" while the converse holds.

You may define a statement of a theory as "true" if it holds in every model, and as "provable" if there is a logical deduction from the axioms of the theory.

Then Goedel's first incompleteness theorem tells us, that any theory, which is as least as powerful as number theory, contains true statements which are not provable.

Hence "true" does not imply "provable" while the converse holds.