Is 就玛丽很快不哭了 grammatically wrong?
When the conjunction 就 means "just" or "then", it doesn't go before noun 

玛丽 is a noun

The sentence should be 

听了我的话，玛丽很快(就)不哭了。 --> Mary quickly (just) stop crying

or 

听了我的话，玛丽(就)很快不哭了 --> Mary (then) quickly stop crying

Edition:

Notice: When 就 means "ONLY", it does go before noun

Example: 他们都哭了，(就)玛丽不哭 --> They all cried, (only) Mary didn't