Yes, this is a modulation to the relative minor, which shares the same key signature. In classical music, although sometimes in popular music too, accidentals would be used to create strong cadences within the minor key. For instance, the leading note of A minor would be raised to G#, particularly when used as part of chord V, to create a major V chord, and so a strong V-I cadence. So, although the key signature doesn't change, often the introduction of raised leading notes (and sometimes sixths, too), can be a clue that you have modulated to the relative minor key (when reading a score, for instance).
When first reading your question title, I thought you might be referring to a move from major to minor while retaining the same tonic note, eg. C Major to C Minor. This is quite different, and is a modulation to the tonic minor. This would require a change of key signature, to three flats in this case, along with (possibly) the characteristic raised leading notes.