Not really. Carter is a qualified expert in some things, but analyzing media bias is emphatically not one of them. Now, as a statement that seems at odds with Carter's assumed partisan bias, him saying it may be more credible as a genuine statement of belief than of a source said the same thing with either no obvious partisan bias or consistent with obvious partisan bias, but that's not the same as lending credibility to the claim itself.
I honestly believe the difference now is Trump itself. He's uniquely unqualified to be president on every intellectual or moral metric I would consider relevant. Therefore, an objective press would be expected to be far more negative than positive.
The other problem (or clever strategy on the right) is to paint mainstream outlets like NY Times or Washington Post as the 'other side', while basically ignoring truly left-leaning media, like Huffington Post or Vox. In doing so, they portray a spectrum where Fox and Breitbart lean to one side, and NY Times / WaPo lean to the other, with the unbiased position somewhere in the center - when in reality, the average of those publications is pretty far on the conservative side of the spectrum. (Even more so for those who actually believe Fox is 'balanced' and other mainstream outlets are in fact far left.)
I don't think it's necessarily impossible to do good journalism while being biased; I think Vox does a fair amount of good reporting for instance. It is necessary though to clearly distinguish between fact and opinion.
But I don't think those are it. When Jimmy Carter says the media is applying a different standard to Trump, it lends credibility to the idea that they just might be.