You're arguing with your girlfriend, who doesn't have a career plan or a good job. She just gets money from her parents. You think she should work hard on her career so that she will be able to support herself. You tell her:
You can't count on someone always being there to support you.
To "count on" someone doing something means that you totally trust that they'll do it. You don't prepare for the possibility that they might not do it.
For example, if a person has spent all her money and doesn't have enough to pay for her rent on the 30th of the month, you could say:
She's counting on getting paid on the 25th.
That means that she strongly believes she'll get paid and doesn't have a plan for what to do if the payment doesn't come.
When the thing you're planning on is a person's action, you say "count on (someone)(doing something)":
Don't count on Sandra being available on Friday. She'll probably be busy then.
When someone "is there to support" you, it means that they are willing to help you (they don't mind helping). It also means that they are able to help you (they can help).
The kind of help or "support" that a person gives can be monetary support, which means that they will pay for you. Another kind of support is emotional support, which means that they will listen to your problems and give helpful advice.
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