Whether we can start a sentence like this with a word “Whether”?

Question

Whether we can start a sentence like this with a word  “Whether”?

Answer

Correction:  Whether we can start a sentence such this one with the word “whether” is contingent on what?

To state it this way still sounds a little funny.  A better way to ask this question is to simply ask:
Can we start a sentence with "whether"? If so, would you provide some examples?

Answer:
Yes!

example:
Whether you like it or not you can start a sentence with 'whether'.

Answer

yes, you can
e.g.  whether to stay or leave is my own business.
whether you agree or not, i'll take her with me.
whether you like it or not, you have to take that test.

Answer

Of course you can start a sentence with weather

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