The spectrum quality problems affected early models of Baofeng HTs and were corrected many years ago. Even the later revisions of the older models, such as the UV-5R, meet FCC purity regulations.
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While I commend the person for being concerned enough to check, there really is no easy way to do it. Aside from needing a spectrum analyzer in the first place, one would have to test with primary signals on a range of different frequencies and then look for spurs well into the microwave region. Nor is it clear that a spur would land on a convenient integral multiple of the primary signal, although that would be most likely, and a real examination would require a thorough process and a laboratory.
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My advice is not to worry about it.
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