You need to check with the unemployment office or on the unemployment website. It is probably possible for you to have federal and state tax withheld from the unemployment checks you are receiving.
In California, which does not tax unemployment benefits, when you certify your work situation every two weeks there is a checkbox where you can request that federal income taxes be withheld. However, as I've seen that they withhold only $3 on a $615 benefit, it is clear to me that their computers are only programmed to withhold federal tax on the $15 state portion and are not withholding for the $600 piece. I have asked my local congressperson's office to contact the CA Employment Development Department (ordinary mortals stand little chance of getting through on the overloaded phone lines to the first level folks and a minuscule chance of then connecting to someone higher up who has the facts) about this and also whether the $600 will be prorated for the last week that starts in July and ends in August or whether there will be no federal benefit for that final week because it ends after July 31.
The news is that the US Department of Labor has stated that no benefits will be paid for the week ending after July 31, 2020. The last $600 payment will be applied to the week ending 7/25 or 7/26, depending upon the individual state's definition of their unemployment week.
If your state unemployment department has not withheld, or underwithheld, income tax on your unemployment compensation (typically 10% federal plus state if your state taxes those benefits), you can make "estimated tax payments" yourself.
On the federal side, such payments are made on form 1040-ES. Whether you need to pay quickly depends upon whether you will end up owing more than a $1000 come next April. You can safely ballpark that need by taking your total untaxed unemployment you've been paid to date this year, adding it to whatever taxed income you have received to date, doubling the total, and using the table on page 7 of the 1040-ES to calculate an amount of tax. Subtract from that what you've already paid in federal tax this year, if any, and compare the result to $1000. If it is more, then it would be safest to send in an estimated tax payment for the difference using the second quarter 1040-ES strip in the form. (The total income figure will likely be an overestimate as the federal unemployment supplement will be replaced by something a good deal less generous, if at all, but better safe than sorry. I wouldn't like you to be stuck with a large interest and penalty bill for significant underpayment.) If you cannot afford that size estimated tax payment, pay what you can afford and try to at least match the 10% that would normally be withheld and which matches the lowest federal income tax bracket rate.
If you care to name your state of residence, I can advise you if you will need to follow a similar procedure for your state.