Not immediately, the body needs time for implantation and hcg to rise. A home test works best after the missed period, roughly two weeks plus after conception. A blood test can pick it up a little earlier.
There is a wait involved, usually around 10-14 days before tests are reliable, after the missed period being clearest. A beta hcg blood test at the clinic detects it earliest. Dont test the next day, it shows nothing.
I always want to know right away honestly but you really cant detect it soon after. It takes about two weeks before a test shows reliably. The waiting is the hardest part.