No, but it is an area of research within linguistics. The
reconstructed Language is called Proto-Human or Proto-World.
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Linguists refer to the first language ever spoken as
Proto-World, but it was spoken long before 6000BCE. It is also
called Proto-Sapiens, and Proto-Human.
The current theory states it was spoken between 100,000 and
200,000 years ago.
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There is no Proto-Indo-European language group.
Proto-Indo-European, or PIE, is the hypothetical root language from
which Indo-European languages today (and others that are extinct)
descend.
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Languages are organsed into families. Languages in a family are
related through descent from a common ancestral language or
parental language, called the proto-language of that family.