Herman (Hermann, Herrmann) and Ármin are the same name in differrent languages, so you should pick a language and use just the one version. If he named his son Béla, he considered himself Hungarian, so I would use Ármin.
Trencsén (note spelling) was the name of both a county and a town within that county in the Kingdom of Hungary. There is no Hermann or Zukmann recorded in the town's Jewish register of births in 1856 (https://www.familysearch.org/ark:/61903/3:1:939F-BTKQ-R?i=18&cc=1554443&cat=1472686), so either the date is wrong or the placename refers to the county. (Or my assumption about their religion based on their names is wrong, but I don't think this is likely.)
The 1898 Budapest address book lists a Hermannn [sic!] Zuckmann, agent, living in the VII. district at Murányi street number 37 (https://library.hungaricana.hu/hu/view/BPLAKCIMJEGYZEK_10_1898/?pg=1210&layout=s). Based on this, I found Béla's birth in the civil registrations:
FS - HCR - PPSKK - BP VII - Births 1898 Aug - 131/255
Number 3549. Entered Nov. 2, 1898, born Oct. 14, 1898.
Child: Béla, male, Jewish
Parents: Ármin Zuckmann, agent, Jewish, age 43, born Nagy-Tapolcsány, Nyitra county; Márta Strauss, Jewish, age 25, born Ludwigshafen, Rheinland [Germany]; residence Budapest VII. Csengery street number 30.
Unfortunately, the Jewish registers for Nagytapolcsány (now Topoľčany, Slovakia) do not appear to have survived; at any rate, they were not microfilmed by FamilySearch.
Edited to add: Jana Shea above pointed out the 1869 census record that may be Ármin and his parents and siblings. Here's the image for it: https://www.familysearch.org/ark:/61903/3:1:3QS7-899K-499M-Q?i=22&wc=QZ7W-SYM%3A323642401%2C324173401&cc=1986782
FS - Slovakia Census 1869 - Nyitra - Nagytapolcsány (Topoľčany) - 23/696 (House-number 9, family 2)
mezzazine floor ("half-upstairs"), 2 rooms, 1 kitchen, serves solely as residence.
Family 2, person 1. Adolf Zuckmann, male, born 1829, Jewish, married, lumber merchant, birthplace N Tapolcsány, local, present, literate
2. Fani his wife, female, 1832, N Bosány
3. Hermann son, male, 1855, single, assistant, N Tapolcsan
4. Malvin daughter, female, 1860, single
5. Náthán son, male, 1864, single, not literate
6. Filip son, male, 1866, single
7. Sofia daughter, female, 1868, single
I think the JewishGen indexer put question marks on Zuckmann because the first letter looks like it may be an F. (The rest of the Fs on the page are lowercase, including the ones in Fani and Filip.)
The wife's birthplace is Nagybossány, Nyitra county, which is Veľké Bošany in Slovakian. Jewish residents were recorded in Nagytapolcsány.