Single member LLC, recently married - Do I file quarterly as "married, jointly" or "married, separately"?

Tax returns are not filed quarterly, only once during the tax year.  Your filing status is not relevant when making an estimated tax payment.

If you are paying estimated taxes you can use either the Form 1040-ES and mail your estimated tax payment to the IRS or you can use the IRS payment website - https://www.irs.gov/payments

When filing your annual tax return you can file as Married Filing Jointly.

You do not file "quarterly taxes", but you might be paying estimated tax on a quarterly basis.   The quarterly estimates you pay do not dictate how you will be filing your tax return.   You want to make sure you are paying enough to avoid an underpayment penalty.   If your spouse is having tax withheld from their paychecks too, then they might want to look at their own W-4 withholding amounts and decide if they want to have more withheld.

  If you were married before the end of 2023, then your filing choices for your 2023 tax return will be married filing jointly or married filing separately.   Joint is almost always better.   

If you have self-employment income, then when you prepare your return you will need to use either online Self-Employed or any version of the CD/download so that you can prepare your own Schedule C for your business expenses.   Your spouse's W-2 can also be entered on that same tax return using the Self-Employed software or the CD/download.

You can make estimated payments on the IRS site:

https://www.irs.gov/payments

If you are legally married at the end of 2023 your filing choices are married filing jointly or married filing separately.

Married Filing Jointly is usually better, even if one spouse had little or no income. When you file a joint return, you and your spouse will get the married filing jointly standard deduction of $27,700 (+$1500 for each spouse 65 or older)  You are eligible for more credits including education credits, earned income credit, child and dependent care credit, and a larger income limit to receive the child tax credit. 

If you choose to file married filing separately, both spouses have to file the same way—either you both itemize or you both use standard deduction. Your tax rate will be higher than on a joint return. Some of the special rules for filing separately include: you cannot get earned income credit, education credits, adoption credits, or deductions for student loan interest. A higher percent of your Social Security benefits may be taxable. Your limit for SALT (state and local taxes and sales tax) will be only $5000 per spouse. In many cases you will not be able to take the child and dependent care credit. The amount you can contribute to a retirement account will be affected. If you live in a community property state, you will be required to provide additional information regarding your spouse’s income. ( Community property states:  AZ, CA, ID, LA, NV, NM, TX, WA, WI)

 If  you are using online TurboTax to prepare your returns, you will need to prepare two separate returns and pay twice.

https://ttlc.intuit.com/questions/1894449-married-filing-jointly-vs-married-filing-separately

https://ttlc.intuit.com/questions/1901162-married-filing-separately-in-community-property-states

https://ttlc.intuit.com/questions/1894449-is-it-better-for-a-married-couple-to-file-jointly-or-separately

Try the Taxcaster tool:

https://turbotax.intuit.com/tax-tools/calculators/taxcaster/

https://www.irs.gov/individuals/tax-withholding-estimator

https://www.irs.gov/pub/irs-pdf/fw4.pdf

https://ttlc.intuit.com/questions/1901110-do-i-need-to-make-estimated-tax-payments-to-the-irs

https://turbotax.intuit.com/tax-tools/calculators/self-employed/