Yes, you should have a conversation with the owners about earning equity.
Before your conversation, you should educate yourself on why and how equity compensation is typically distributed in new ventures. Fortune has a decent primer, but make full use of Google and have some conversations with people with startup experience.
A brief rundown of what you ought to know going into the conversation:
1. Why would your company choose to compensate you with equity? Cash compensation is cheaper to the company and can have all of the same retention incentives as equity. Know that paying you a "dollar" in equity will cost your company more than paying you a dollar in cash. The biggest reason that early companies choose to distribute equity as compensation is simply because they don't have enough cash - so it is a good alternative to asking for a cash raise in a new venture.
2. What is the total value per year in equity or equity-equivalent compensation you expect? How much more do you think you should be earning per year or per quarter? Your equity compensation is likely to vest over time, so this would be the vesting rate you're targeting.
3. What conditions are you willing to accept? You equity compensation will likely vest according to conditions (e.g., time in the company and/or completion of specific milestones). You may even be required to stay with the company through an IPO to be able to exercise distributed options. Know what you are willing to sign on for.
4. What types of compensation are you willing to accept? Equity compensation is rarely distributed to employees as common stock. You are more likely to receive options or, in some cases, restricted shares. Different equity and equity-like compensation models have different mechanics - figure out which ones you think match your financial expectations. Greater liquidity, flexibility, and preference all come at a cost.
5. How will your compensation perform in liquidity events (e.g., IPOs) or future rounds of funding? You should know whether or not the compensation you are asking for has features like dilution protection. This is really an extension of the point above, but these are the events that allow you to "cash out," so you really need to understand how they work to be able to calculate the present value of any equity compensation.
Good luck in the conversation! You should always negotiate for better compensation if you feel you are underpaid - you're on the right track!