YES, this is a scam, google "ftc.gov advance fee scam" for more info. This is a version of the advance-fee-for-something scam where something is a grant, loan, lottery winnings, interest rate reduction, gift card, doggie, cruise, vacation, free gas, etc., and the advance-fee is for tax, shipping, registration, secure handling, service, etc. They may send you a fake check, may want your credit/debit card or bank account number,  or tell you to wire cash via Western Union, Moneygram, etc. They can put recurrent difficult-to-cancel monthly charges or a 1-time charge on your card or bank account, and sell account info to other criminals and you'll have to replace accounts altogether. Wired-cash is irretrievable - once it's picked up at the other end, it's untraceable and gone. If they want you to call them up for instructions, they may tell you not to discuss the wire with Western Union desk clerks but the real reason is WU clerks are familiar with these scams and will warn you not to wire the money.
For more info, google "ftc.gov advance fee scam." Report to US Postal Service Inspectors https://postalinspectors.uspis.gov/forms/MailFraudComplaint.aspx , the Federal Communications Commission http://esupport.fcc.gov/complaints.htm , the Federal Trade Commission https://www.ftccomplaintassistant.gov/ and your state AG, as appropriate.

Gentlemen from Global Customer Awards called me mother and told her she had won $35,000 and then asked her to call Western Union phone number  800-325-6000 to verify his legitamacy.

Is it a scam?