Looking for one of his other sisters certainly seems like a good idea.
I first found Anton Martinus Nielsen in the IGI, born 1860 in Aarhus, Denmark along with four brothers but not a sister in sight

But the baptism in 1860 and the place of baptism Aarhus, tallied with information on his records from the ship Friedeberg on which he travels to Brisbane, Australia - departing Hamburg 22 June 1878 aged 18, previous place of residence Aarhus and country Denmark.
Here are the four boys with their parents from the IGI: Parents: Niels Andersen Moller and Mette Jensdr and their sons baptised in Storring: Anders 1856 / Jens 1858 / Anton Martinus 1860 / Jens Christian 1863 - all with the surname Nielsen.
Later on I came across a genealogy website in Denmark which seemed to indicate that this couple (Niels Anderson Moller and Mette Jensdatter) had other children born prior to those listed, it included Anders and Jens but not the two younger boys. Here are the names included on that website:
Andersen, Ane Sofie 1849 (there's a christening for Ane Sophie Andersen on the IGI 18 Nov 1849)
Andersen, Ane Mette b. 15 Nov 1851 (christening on the IGI 11 Jan 1852)
Andersen, Jensine b. 10 Jan 1854 (christening on the IGI 4 May 1854)
Nielsen, Anders b. 27 Nov 1855 (christening on IGI 1 June 1856)
Nielsen, Jens b. 25 Apr 1858 (christening on IGI 6 June 1858)
This website indicated that Niels Andersen Moller dies 11 Sep 1864 and following her husband's death, Mette Jensdatter marries Soren Frandsen on 11 Nov 1865. Frederikke Frandsen b. 4 Apr 1869 is the only child of this marriage noted here on this webpage and she's the Freda/Frieda Frandsen/Frantzen I was hoping to find in Richmond Valley Staten Island in 1928.
I guess I'd better double check that website and make sure there are no other daughters from the second marriage! Otherwise I'm looking for Ane Sofie Andersen, Ane Mette Andersen or Jensine Andersen who married a Mr Matson.
Cheers
Mumfy