Generally, converting from a pre-tax account to a Roth account is considered income and is taxed in the year the conversion occurs.
When a conversion is made, a 1099-R form will be issued for that tax year by the financial institution that processed the conversion. Box 7 on the form will generally have a code "2" (if under age 59 1/2) or "7" (if over age 59 1/2), which indicates a distribution has occurred.
This article is a good resource for Roth IRA conversions: https://blog.turbotax.intuit.com/taxes-101/taxes-101-ira-conversions-1946/
Are the conversion taxes payed through the IRS estimated tax process or do we pay the taxes when we file our tax return?
The conversion is considered ordinary income, and would generally be subject to estimated tax rules. Waiting until the return is filed to pay any tax related to the conversion may incur under-withholding penalties.
Plan administrators/financial institutions also can withhold taxes during the conversion, although this would reduce the amount that ends up in the Roth account after the conversion.