Thanks to Cando I have now checked Qld ... I didn't notice that in the heading oops !
I can see online index registration in Qld for 4 March 1889 for Eleanor Sarah NELSON, dau of Charles and Agnes (Needham)... so both Agnes and Charles would have needed to be in Qld together by mid 1888 .... how sure are you that the shipping records show a son Charles aged 2 with Agnes arriving Feb 1888 ... I am wondering if the list shows this lad as her son... perhaps it is her husband aged in his twenties ... Eleanor's b.c. should show if she had older siblings. And Charles 1893 d.c. should show all the children of the marriage with Agnes.
tip, when using the date range, at the link that Cando has given you, once you know the year, then search limiting to one half of the year, then one quarter, then the month ... etc not hard to then find the exact date.
I note that Frederick was registered as John Frederick. I have not checked for his exact d o b ... I have not checked for Charles exact d o d ... but these options are available at the Qld BDM free to search online index as linked above by Cando at #2..
Fingers crossed this helps ..
Cheers, JM