No, not normally.
Most insurance companies offer insurance terms of only six or 12 months --but the good news is that you don't have to pay for the policy in full at inception.
Typically, you can make a down payment then make monthly payments on you car insurance policy. Each state has requirements on the minimum down payment, but it is typically 25 percent of the full policy premium.
At CarInsurance.com you can purchase insurance for six months or 12 month, but you have the option to cancel your policy when you don't need the insurance anymore. Some state laws may restrict you to maintain the policy for at least 60 days without reason to cancel.
If you need insurance for just one month, you can purchase insurance for the shortest term offered and request that it be cancelled after one month. In most cases, you will need to make this request in writing. It is also possible that you will need to provide additional information and may be subject to an early cancellation fee if you cancel mid term. Some states have pro-rate cancellation refunding setup.
Some states with mandatory insurance laws require proof of other coverage or proof that there is no insurable interest (you sold the car, moved it out of state, destroyed it, etc.), in order to cancel within a certain amount of time (typically 60 days).
We are happy to sell you a state car insurance policy and to accommodate your needs.
Comment Update: There is not a specific company or site that offers one month policies. Typically, the shortest term is six months, but you have the opportunity to cancel the policy. We don't currently offer one month policies in any states, but some of our out-of-site partners may offer them.