A spokesperson for the Gulf carrier said it would only carry passengers to the US who had the correct documentation.
"We are enforcing the new rules," said the official.
"If travellers to the US don't have the proper documentation, we are not going to take them to the US."
The news comes after Trump imposed tough new controls on travellers from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen.
The newly inaugurated US president signed a sweeping executive order yesterday to suspend refugee arrivals and impose tough controls on travellers from the seven countries.
These included the permanent resident card, known as the "Green Card", and up to eight different types of visa.
Qatar Airways destinations in the US include New York, Atlanta and Chicago.
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