No, not correct.

"anytime soon" (like all "any" structures of this type) belongs in questions, negatives and conditionals.

"sometime soon" would do. Or "before too long".

#1 addition... in the foreseeable future

btw start a family = have a child

Sandra, geht das "haben" im Deutschen nicht am Ende des Satzes? ... Familie in absehbarer Zeit haben.

I want to have my own little family soon.
My own little family is in the making.
My own little family is on the horizon.

;-)

"anytime" = nee
"sometime" = ja