Galway is always busy in summer. Good chance of a place in Salthill which might be a good idea as it is away from the noise of Quay Street - especaillly when you have a baby. Salthill is well connected with a good bus service from to bottom of Quay Street or Eyre Square.
I suggest:
Galway Bay Hotel, Salthill,
Salthill Hotel,
Warwick Hotel, Salthill.
In the city centre,
Victoria Hotel,
Imperial Hotel,
Skeffington Arms Hotel,
Park House Hotel
Meyrick Hotel
..........all of the above are in or around Eyre Square which is 500 metres from Quay Street.