No, the Cult depicted is an entirely new one, as has been affirmed by director Christophe Gans:
There isn't just one Cult in Silent Hill. Why? Because what we wanted to do with the movie, and we probably were a bit ambitious, was to open the Silent Hill mythology, to open as many leads as possible.
The main difference between the film and the games is the Cult and its leadership. The cult, known as the Order, believes in a god they call "The Holy Mother" or "The Lord of Serpents and Reeds". Their opponents called this deity "Samael". In the games, it is Dahlia Gillespie who burns her daughter Alessa alive as part of a ritual to bring about the god's reincarnation on Earth. She chooses Alessa because, at a young age, she exhibits extreme psychic powers. In Silent Hill 3
, it is revealed that "she could kill people with her mind." The ritual goes awry, and Alessa barely clings to life with only the half of the god's soul is reborn inside her. As a result, half of Alessa's soul is reincarnated as a baby later found and adopted by Harry Mason, and named "Cheryl".
On the brink of death, Alessa is kept alive unnaturally by her mother, in hopes that her pain and suffering will attract the other half of the god's soul. She is kept alive secretly in the hospital basement due to the efforts of Dr. Kauffman, Dahlia's partner in a drug-running business that also helps to fund the Order's activities. Though she has been previously led to believe she will be released from the hospital after her convalescence, Alessa realizes the extent of Dahlia's plot 7 years later. Determined to prevent the god's birth, she summons Cheryl, her other half, back to the town so that once fully reunified, she can manifest her powers, and kill herself. Harry, unaware of all of this, tries to find and save his daughter who has become lost in Silent Hill (or so he thinks), while Alessa tries to prevent him from discovering the truth throughout the course of the game.
In the end it is revealed that this reunion was inevitable, and moreover, was monitored by Dahlia herself, who had been using Harry to methodically seal Alessa's powers. Dahlia actually succeeds in raising the god who ironically kills Dahlia instead of rewarding her. However, having been born from this botched ritual, the god is weak and is killed by Harry. As Alessa lays dying, she is once again reborn as an infant, whose care she again entrusts to Harry, having proven himself a worth parent. This baby later grows up to be known as Heather, the lead protagonist of Silent Hill 3
, the game that is the direct sequel to these events. See here
for more information.