Does DESeq2 work for investigating the gene differential expression before and after the treatment but in the same sample? like when we used paired sample t-test that the samples are the same but before and after for example treatment?

I saw that people used condition and time for the design, but in my data time is not specified.

Does DESeq2 work in this situation?

Thanks a lot for your kind help in advance.

Yes, paired samples are supported. Please see the vignette and one of many posts on 'deseq2 paired' via google.