I think the solution here is to use your continuous covariate as continuous, rather than categorizing it. Categorizing continuous covariates is commonly done; it may make interpretation easier for clinicians, but statistically it is always a poor choice.

There seems to be some concern here about whether or not to convert continuous variables into variables with just two (or more) categories. Let me address that here, rather than in a comment. I would keep all of your variables as continuous. There are several reasons to avoid categorizing continuous variables:

• By categorizing you would be throwing information away--some observations are further from the dividing line & others are closer to it, but they're treated as though they were the same. In science, our goal is to gather more and better information and to better organize and integrate that information. Throwing information away is simply antithetical to good science in my oppinion;
• You tend to lose statistical power a;
• You lose the ability to detect non-linear relationships;
• What if someone reads your work & wonders what would happen if we drew the line b/t categories in a different place? (For example, consider your BMI example, what if someone else 10 years from now, based on what's happening in the literature at that time, wants to also know about people who are underweight and those who are morbidly obese?) They would simply be out of luck, but if you keep everything in its original form, each reader can assess their own preferred categorization scheme;
• There are rarely 'bright lines' in nature, and so by categorizing you fail to reflect the situation under study as it really is. If you are concerned that there may be an actual bright line at some point for a-priori theoretical reasons, you could fit a spline to assess this. Imagine a variable, X, that runs from 0 to 1, and you think the relationship between this variable and a response variable suddenly and fundamentally changes at .7, then you create a new variable (called a spline) like this:

  XsplineXspline=0=X−.7if X≤.7if X>.7

  
  then add this new Xspline variable to your model in addition to your original X variable. The model output will show a sharp break at .7, and you can assess whether this enhances our understanding of the data.
  

Well, what you're doing wrong is using as the reference group a group with zero events. Instead of hazard ratios, think in simpler terms (in my opinion) of incident rate ratios (IRRs), where the incident rate (IR) is IR=number of cases / total person-time.

IRRquartile 4 vs. quartile 1=IRquartile 4IRquartile 1

What happens if IRquartile 1=0?

You can change your categorisation (use tertiles or some other meaningful categorisation) or, even better, if you have a continuous predictor you can treat it as such and examine potential nonlinear relationships using polynomial terms, fractional polynomials or restricted cubic splines, for example.

The hazard of an event is the instantaneous probability of an event occurring at time t, conditional on it not having previously occurred.

Your problem should be clear instantly - with no events, the probability is zero. Borrowing from andrea's example, the incident rate is equivalent to a constant hazard - in your case, a constant hazard of zero.

Dividing by zero tends to make software angry.

You need to switch your reference category. My suggestion is to use "Quartile 4" or the other high value of the category, and step down, rather than using Quartile 1 and stepping up. If you were hoping to, for example, show an increase in the HR as you moved up a category, you're now showing the equivalent protective effect from moving down one.

I would also suggest taking a moment to consider why you have no events.
It's possible you're simply having a run of "bad luck", at which point there's nothing you can do but increase the study size or follow the population for longer in hopes of accumulating more events. But you should make sure there's no reason that the probability of having an outcome in your population isn't zero for a reason. For cardiac events I can imagine one, but it is always worth stopping to consider when you have zero events in some level of a covariate.