The G-4 spouse is an "exempt individual" (exempt from the Substantial Presence Test), which would make the G-4 spouse a nonresident alien.
The couple can each file as Married Filing Separately, or, they can choose to treat the G-4 spouse as a resident alien and file as Married Filing Jointly, using the Nonresident Spouse Treated as Resident election.
I think that the income will remain non-taxable. The G-4 spouse income is currently exempt from tax either due to a tax treaty or a general provision of US law. Tax treaty provisions sometimes also apply to resident aliens; you would need to look at the exceptions to the "saving clause" of the specific tax treaty to make sure. Even if a tax treaty doesn't apply, generally non-US-citizen, non-permanent-resident employees of foreign governments are not taxed on their salary, regardless of whether they are resident alien or nonresident alien, as long as they perform similar work to what US government employees perform in that country, and the same exemption is granted to those employees.