Your first suggestion, 'We are close in age', is fine, but not very common as an expression. You would by most likely to hear this in reference to siblings e.g. 'Her three children are very close in age.' The second isn't quite right. It should be 'We are of a similar age.' More informal alternatives would be variations on this sentence: 'We are ........... the same age'. In the gap, you could put a number of different modifiers, for example: almost, nearly, about, around.

Both sentences are correct, but in everyday speech I would say "we're almost the same age."

I would say, "you and I are close in age." Or, "you and I are almost the same in age."