What do the letters L and G stand for? My guess is a Latin phrase but I cannot think of what phrase relates to Liège.
The official site of the city of Liège states that
The letters L and G on the coat-of-arms correspond to the Latin expression "Libertas Gentis", which means "Freedom to the People".
Interestingly, the French Wikipedia article about perrons says that originally the letters were added to distinguish the coat-of-arms of Liège from that of Saint-Trond, which also featured the perron and the letters S and T, and that in the Middle Ages, the coat-of-arms carried the syllables Lie and Ge or Ly and Ge (which seem demonstrated by the picture in Thomo's answer); it also goes on to add, that the "Libertas Gentis" meaning is a recent legend the people from Liège are fond of — they very probably are, since that's what the Liège website says!
Further confirmation is provided by a website (in French) that cites a text by Liègian historian Théodore Gobert:
Could we take seriously, for example, Ferdinand Henaux's claim that the LG letters "are the initials of the words Libertas Gentis (11), given that he does not provide a single text to support it? In the 17th century, Foullon glimpsed the truth when he noted the belief that these letters "mean Légia or Liege (12). It is actually an abreviation, the radicals of the two syllables that make up the name of the city. Thus in Sint-Truiden, the perron is flanked by the letters S and T. Likewise, the word Lie-Ge appears, cut by the old national emblem, in vignettes from the oldest publications from Liège. The same word Ly-Ge is found, cut by the perron, on a fireplace in the former monastery of St. Lawrence, crafted in the first quarter of the 16th century (13). So, the letters were there just to distinguish the perron of Liege from that from other localities.
So it seems they are just the initials of each syllable in the name Liège.
It would appear that it is a shortening of Liege to LG.
According to the Heraldry Wiki page for it, the word Liege originally appeared either side of the perron:
Ever since the perron, including the base with the three lions, has been the arms of the city. The actual shape, however has varied widely during the centuries, and similarly, not all images show the lions. In the late 17th century the whole name, LIÈGE was shown around the perron. The letters L and G appear for the first time in the late 18th century.
I found this image, presumably with the 17th Century version.