According to Black the name did originate with the burgh of Glasgow. It is an old name with one of the earliest known being John de Glasgu in 1258.
Mac Giolla-Domhnaigh in his book says the this name in Ulster is an Anglicized for of McCluskey.

The Surnames Database has this entry:
This long-established surname is of early medieval Scottish origin. It is a locational name from the old burgh, now city, of Glasgow on the River Clyde, first recorded as "Glasgu" in 1116. It is believed to be named from the Ancient British words "glas", meaning grey or perhaps green, and "cau", a hollow. Situated in west central Scotland, Glasgow is the administrative centre of the Strathclyde region, and is the largest city in Scotland. The city contains the cathedral of St. Mungo, dating from 1179, whilst its university dates from 1450. The Exchequer Rolls of Scotland record that in the year 1289 Andrew de Glasgow, an official of the burgh, was appointed to take notice of the "escheats" or forfeitures in the county, and to certify them to the Exchequer. Another early recording is that of John of Glasgow, given as being a monk of Holyrood in 1299; and in 1343, payment was made to John de Glasgu for the table for the king's servants. Robert Glasgow, noted in "Protocols of the town clerks of Glasgow" was a witness in the burgh in 1554. A coat of arms granted to the Glasgow family of Scotland has the blazon of a silver shield with an azure chevron between two black fishes naiant in chief, and an oak tree growing out of a green mount in base. In both Ireland and Scotland, the surname Glasgow is occasionally found as a developed form of the Gaelic "MacCluskey", originally "MacBhloscaidhe", and a patronymic of "Bloscadh" meaning "Loud One". The first recorded spelling of the family name is shown to be that of John de Glasgu, the bishop of St. Andrews.This was dated 1258, in the "Charter Book of the Priory of St. Andrew's", during the reign of King Alexander 111 of Scotland, 1249 - 1286. Throughout the centuries, surnames in every country have continued to "develop," often leading to astonishing variants of the original spelling.

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