Yes, you can. It is not needed in a scenario you described, but can be useful if, for example, you want different mixes of investments or different beneficiaries or something like that.
For the purpose of contribution limits or conversion calculations all your IRA accounts in all the brokerages are lumped together and considered one. So there's no benefit in having a special traditional IRA account just for conversion to Roth when you already have a traditional IRA account. You need to remember that the conversion will be prorated based on the balances in all your traditional IRA accounts.
Let's use some example. In the comments you suggested that you contributed $5K last year, which were used for a tax deduction, and you want to do a backdoor conversion for another $5K this year.
You'll have 10K in traditional IRA of which you converted 5K, so the converted 5K would be prorated. If the second contribution was after tax then it will not be taxable, but the first contribution was pre-tax, so it would be. In this case you'll have to recognize 2.5K taxable income (50% of the conversion, since 50% of the balance was pre-tax). It would be best to completely empty all of your traditional IRAs for this to work most efficiently.
I wrote a detailed answer including describing the technicalities of the reporting, to a similar question here.