Argos had slipped away from his friends under the bridge, feeling guilty for leaving them but not able to stay - the lure that Odysseus may well come to see his old friend was too much for him to resist and he made the choice to accept the lady's very kind offer. He was still amazed that any woman would offer such a kindness to him, and although there maybe should have been some suspicion about her reason for doing so, there wasn't. Argos was a dog after all, kick him and he only came crawling back later.
He was sat in a little huddle on the ground, in person form because being a dog and running loose around Central Park tended to attract too much attention. He looked up when he heard his name being called and smiled, would have wagged his tail had he still had one, and stood up to greet the lady. "Hi! Thank you so, so much for letting me stay at yours, really, I don't know how I'll pay you back but I will, promise!"