I'm currently in the military and stationed in Colorado. I'm a Washington State resident.
My wife is a Colorado State resident. Her only income was a 1099-MISC. Do we need to file state taxes with Colorado?

Normally, income earned while in Colorado is reported to Colorado.

If you are present in a state, then any income earned during this period, must be reported to the state.
To your resident state, you must report your worldwide income for the year. Read: Your U.S. Tax Return: Working or Living in Two or More states

About 1099-misc: You will report your income and expenses on schedule C or C-EZ (Form 1040). This income is subject to SE tax at 15.3%. Read: Your U.S. Tax Return: Tax Filing by Self Employed Sole Proprietor or Independent Contractor

Military are EXEMPT from state income taxes in the states where they are stationed. Since Washington has NO state income tax, your military income has NO state income tax liability at all.

Your wife's income, on the other hand, is liable for Colorado state income taxes. When you are transferred from Colorado, she will assume your state domicile and her home state tax liability will be Washington. However, if she gets a job in your next duty location and that state has a state income, she will pay taxes as a non-resident who is earning income within that state's borders.