We want to know if me the father and mother, can change the surname of our daughter age 6 from mother to father surname on birth certificate. As the father I was at the register of birth my detail are on and singed by me on the birth certificate.

Maybe. Depends on the laws of wherever the child was born.

It was England Lincoln place of daughter of birth

it was England Lincoln place of daughter of birth

Read all about it here. (http://www.direct.gov.uk/en/Governmentcitizensandrights/Registeringlifeevents/Birthandadoptionrecords/Registeringorchangingabirthrecord/DG_175618)

"... You can only change a child's surname in their original birth record to correct a spelling mistake.

However, a child can be re-registered in the following circumstances:
•to record the natural father's details in the birth record
•following the marriage of the natural parents, if they were not married at the time of the birth

In these circumstances, you are able to change the child's surname to the mother's or the father's – or a combination of the two – as long as both parents agree to the change. If the child is 16 years of age or over, his or her written consent must be obtained before the surname can be changed.
... "

So, as I read it, you and the other parent will have to marry before the surname can be changed to that of the father.