Inspired by this recent question, "Are the Maesters on the Wall always Members of the Nights Watch" and Jaime's speech on Oaths, do the Night's Watch Oaths conflict with those of the Maester's of the Citadel?
It would appear not, in fact they are quite similar.
Night's Watch Oath
Night gathers, and now my watch begins. It shall not end until my death. I shall take no wife, hold no lands3, father no children4. I shall wear no crowns and win no glory1. I shall live and die at my post. I am the sword in the darkness. I am the watcher on the walls. I am the fire that burns against the cold, the light that brings the dawn, the horn that wakes the sleepers, the shield that guards the realms of men2. I pledge my life and honor to the Night's Watch, for this night and all the nights to come.
There's no specific mention to what the maesters vows are but the below are what I've found about them.
1. Put aside House name
A toothless smile quivered on the ancient lips. "Only a maester of the Citadel, bound in service to Castle Black and the Night's Watch. In my order, we put aside our house names when we take our vows and don the collar." The old man touched the maester's chain that hung loosely around his thin, fleshless neck.
A Game of Thrones, Jon VIII
2. Serve the realm
"He told me that a maester's collar is made of chain to remind him that he is sworn to serve," Jon said, remembering. "I asked why each link was a different metal. A silver chain would look much finer with his grey robes, I said. Maester Luwin laughed. A maester forges his chain with study, he told me. The different metals are each a different kind of learning, gold for the study of money and accounts, silver for healing, iron for warcraft. And he said there were other meanings as well. The collar is supposed to remind a maester of the realm he serves, isn't that so? Lords are gold and knights steel, but two links can't make a chain. You also need silver and iron and lead, tin and copper and bronze and all the rest, and those are farmers and smiths and merchants and the like. A chain needs all sorts of metals, and a land needs all sorts of people."
A Game of Thrones, Jon V
3. Hold no lands or lordships
Grand Maester Pycelle did not disappoint her. "Lord Qyburn?" he managed, purpling. "Your Grace, this . . . a maester swears sacred vows, to hold no lands or lordships . . ."
A Feast for Crows, Cersei IV
4. Father no children
When a maester donned his collar, he put aside the hope of children, yet Cressen had oft felt a father nonetheless. Robert, Stannis, Renly . . . three sons he had raised after the angry sea claimed Lord Steffon. Had he done so ill that now he must watch one kill the other? He could not allow it, would not allow it.
A Clash of Kings, Prologue
Also just like the Night's Watch Acolytes and Novices can leave the order before they become a Maester.
That was many years ago, to be sure. The boy of sixteen was a man past forty now, and his legend had grown a deal darker. He had traveled in the Free Cities, learning the poisoner's trade and perhaps arts darker still, if rumors could be believed. He had studied at the Citadel, going so far as to forge six links of a maester's chain before he grew bored. He had soldiered in the Disputed Lands across the narrow sea, riding with the Second Sons for a time before forming his own company. His tourneys, his battles, his duels, his horses, his carnality . . . it was said that he bedded men and women both, and had begotten bastard girls all over Dorne. The sand snakes, men called his daughters. So far as Tyrion had heard, Prince Oberyn had never fathered a son.
A Storm of Swords, Tyrion V
Note though that the Citadel has been known to kick a maester out of the order but this still wouldn't conflict with the Night's Watch oath.
"The Citadel took my chain." Qyburn put away his needle. "I should do something about that wound above your eye as well. The flesh is badly inflamed."
A Storm of Swords, Jaime IV
There appear to be no contradictions between the vows as far as I can find. In fact, they both seem to align in a number of ways (setting aside house name, celibacy, renouncing claim to titles or wealth). Beyond that, Maesters are expected to make vows and remain loyal to the castles and lords that they serve, which fits in with serving the castles and lords of the Night Watch.