Can I transfer the property I received as a beneficiary of a trust directly to my own trust in California?

Generally it will go to you first and then you can transfer it to your trust. Some trustees will transfer it to your trust directly.

Probably you can assign your interest in the trust into your own trust.

Yes, you can take the property directly into your own trust. If you inherited the property already, or about to, be sure you obtain an appraisal at the actual time of inheritance prior to transfer into trust so you will know what the basis is going into the trust and can file the notice of no change in value at the time of transfer into your trust. You can also take title at the time of inheritance from the estate by grant deed and then transfer into the trust. I would suggest you consult a CPA/Tax lawyer re the tax consequence of receipt either way.

The property should go to the named beneficiary, you. There may be exemptions from a reassessment as a result of the transfer or the property will be appraised at fair market value for tax purposes. After that transaction, you can transfer the property into your trust, which is an exempt transfer. It is best to do transfers one step at a time.

It's best to do by deed to you and then separate deed to your trust. That way you can file your parent-child exclusion (if there is one). Just a cleaner chain of title in my opinion.