No, because the BOS destroyed NCR's gold reserves, forcing them to use paper money. I doubt these reserves were in the Mojave. Likewise, the Mojave Chapter only did battle with NCR at Helios One. There's no indication they did war with NCR anywhere else.
(Helios One for the Brotherhood was a disastrous battle where the Mojave chapter inflicted heavy casualties but lost in the end - For NCR it was Tuesday)
Likewise, I doubt Kimball was referring to the Battle of Helios One as the "Brotherhood War." In my game, Colonel Moore said, "Do you have any idea how long we've been fighting them?" When you bring them the question of a truce with the BOS. For me, that implies there's a lengthier war than the conflict at Helios One - which was again the only real action the Mojave Chapter saw against NCR.
I don't trust the Vault wiki completely but it mentions Moore said she served "Four tours against the brotherhood." Given the length of military terms, it's entirely possible that the war lasted upwards of a couple of decades.
If I had to guess what happened, it's entirely possible that the Brotherhood and NCR's relations deteriorated over the increasing access to weapons technology that NCR undoubtedly had in the aftermath of the Enclave's destruction. If NCR intended to make use of energy weapons and power armor, the Brotherhood might have decided to do a pre-emptive strike (or vice versa). With President Tandi's death VERY likely after the events of the Enclave's destruction, it's possible that NCR reacted rashly or the Brotherhood thought they'd get a quick victory.
Hell, given NCR's history of expansionism, it's believable they might have INSISTED on the Brotherhood/Lost Hills becoming a part of NCR's nation-state as opposed to being a nation INSIDE NCR.
Wars have been started over a lot less, after all.