It depends on where your strengths are, and where you want to go.
You can start with objects, and do a pretty good job, including good table design in your database, if you start in the right place.
Start with object oriented analysis, rather than object oriented design. I can't emphasize the difference between analysis and design strongly enough. People who write about about object oriented analysis, some of whom use UML as a tool, try to make this distinction clearly. Analysis pertains to the problem domain, while design pertains to the solution domain. These can easily be mixed up with each other, whether your approach is object oriented or not.
If you do a good OOA analysis of the requirements of your project, you can probably do a reasonable job of constructing a conceptual data model in parallel, and keeping those two in synch with each other. When you build a conceptual data model, I suggest you keep to a model like the ER model.
An ER model won't tell you how to design your database. That's the whole point. The way you separate analysis concerns from design concerns in data modeling is to do your analysis using ER modeling and your design using relational data modeling, at least at the beginning.
The mapping between OOA and ER is simple enough so that you can manage the two models in parallel. There may be newer tools that manage both kinds of models for you. The mapping between ER and RDM is deceptively simple. In the simplest mapping, you turn each entity into a table keyed on the entity's identity, and each relationship into a separate table keyed on foreign keys that refer to the entities. It's possible and desirable to reduce the number of tables by plugging some foreign keys into entity tables, but that's a detail.
From OOA, you proceed to OOD for design and OOP for programming.
From ER for conceptual data modeling, you proceed to RDM for logical data modeling and to your DBMS specific dialect of SQL for physical data modeling. There are definitely tools to help you with this, if your project is too large for modeling on paper or whiteboards.
Periodically, you build what you've got, maybe with stubs for the unbuilt part, and you
reconcile your application design with your database design. If you do a good job, at the end of the day, you should be in pretty good shape.
If you are designing one database and several applications at once, things get really interesting.