In my tree I have 28 people who were born at 12 Claremont Terrace, Glasgow between 1912 and 1966. One of them records the address as Park Nursing Home, 12 Claremont Terrace.
Also 7 born at 6 Claremont Terrace between 1921 and 1959 and 5 at 3 Claremont Terrace between 1930 and 1938.
The 1935 Valuation Roll lists nursing homes at Nos 1 (Proprietor Henrietta Gunn), 4 (Elizabeth Burns), 6 (Annie L Macpherson), and 12 (Elizabeth M Allan)
No 3 isn't listed at all in 1935, but it is listed as a nursing home in 1930, proprietor Elizabeth Burns.
So it looks as if you have four to choose from.