It's possible, but probably very difficult if you voluntarily resign.
As you may already know, the reason you are out of work affects your eligibility for unemployment benefits. If you quit, were fired, or left work because of a trade dispute, you will be scheduled to a telephone interview because there is a separation issue that must be resolved. The Department interviewer obtains and documents information about the separation from the employer and you (the claimant) and decides, according to law and regulations, if you are eligible to collect benefits.
Typically, when you resign you are not eligible for unemployment insurance benefits. There are exceptions such as when you are forced to resign due to circumstances in the workplace, such as illegal conduct, retaliation, harassment, etc., which make it difficult or impossible for you to continue working there. Without those kinds of circumstances you are unlikely to prevail on a challenge to your eligibility for UI benefits, assuming the employer will mount a challenge. However, you should weigh the value of the UI benefits against having a termination on your work history. Consider that having a termination on your work history may harm your future employment prospects.
Disclaimer: This response does not constitute legal advice. It is merely an opinion, and not intended to create, and does not create an attorney-client relationship. It is impossible to provide a legal opinion in response to your question without a comprehensive analysis of all facts and circumstances relating to the legal issues.
When you're given the choice to quit or be fired and you choose to quit, this is called a "forced resignation in lieu of discharge." The California Unemployment Insurance Appeals Board unambiguously holds that, under these circumstances, the employer is the moving party behind the termination. This alone, however, is insufficient to guarantee your eligibility for unemployment benefits, as the relevant question then becomes whether the employer terminated your employment for reasons of misconduct related to the work (pursuant to California Unemployment Insurance Code section 1256).
While this will always require a detailed factual inquiry with your attorney, please keep in mind that the employer has the burden of proving misconduct in 1256 cases.
I've attached a link to the CUIAB Precedent Benefit Decisions page. Scroll down to "Forced resignation in lieu of discharge" under the "Moving Party" section to read the actual decisions. These may help in the event you choose not to hire an experienced unemployment appeals attorney.