Rather than answer your question directly, this answer will outline a research process for finding the records, using Nashua as an example, so anyone with other towns in New Hampshire can use it as a model for their own searches.
Determine when and where records might have been created
Check the historical statutes for each locality to find the date that the poor farm or poor house was established for each locality. See Kimberly Powell's guide What Was the Law? Historical U.S. Statutes Online.
The description of a talk “Cared for at the public expense”: A brief history of New Hampshire’s Poor Farms and State Institutions, hosted by the New Hampshire Alliance for Preservation in 2018, says that towns in New Hampshire began creating poor farms "[s]tarting in the 1820s".
The New England Historical Society's page A Poorhouse in Each New England State says:
Unlike the other states, which made their towns care for the poor, New Hampshire by 1866 required the county to provide for them. Each of New Hampshire’s 10 counties had a county poorhouse.
FamilySearch Research Wiki
The FamilySearch Research Wiki article New Hampshire Town Records says that town records may include "care of the poor". You may have to look for general collections of town records, rather than a database with "poor records" in the title. The article has an overview of what might be available on microfilm, and suggests other sources for informatiion such as historical societies and town historians.
The Wiki also has a useful overview of New Hampshire Statewide Online Genealogy Records, with a link to tips on Locating Online Databases.
Unfortunately as I write this (24 Dec 2020), the article Nashua, Hillsborough County, New Hampshire Genealogy is mostly an outline with very little information, though it does have the address of the City Clerk and a link to the town's website.
FamilySearch Catalog
Do a place search in the FamilySearch catalog, and a keyword search for each jurisdiction. See Two Hidden Secrets to Find a Ton More Results from the FamilySearch Catalog from The Ancestor Hunt for tips and screenshots.
Using the keyword search "New Hampshire Poor" doesn't turn up any obvious poor records for Nashua in the listings, but you can mine the titles in the search results for further Google Searches.
Do a place search for Nashua and read the catalog entry and Film Notes for any town records to see if they mention records about the poor. Also note anything listed under Inventories, Registers, and Catalogs such as this Catalogue of New Hampshire town records. This film has the camera and key icon, so you'll have to view it at a FHC or at an affiliate library once we have access again. However, any time you have an exact title for any microfilm or book from the Family History Library, you can search for copies online or from other libraries using Worldcat.org.
Under town records for Nashua, the Catalog has Town records, 1639-1840 which does not cover the date range you're looking for.
The collection New Hampshire, Town Clerk, Vital and Town Records, 1636-1947 does not include a waypoint for Nashua under Hillsborough County.
New Hampshire State Archives
Archival Holdings lists several categories of records. Under Town Records, there are two entries, Municipal Records (not online) and Road and Highway Layout Records (some are online). Check the research guides on that page, the general guide for researching at the archives, and the genealogy guide. Search your county and town of interest for local archives and repeat the process.
New Hampshire State Library
The Town Records Index gives the following data for records for Nashua held at the library:
NASHUA
County: Hillsborough
Dates for Town Report: [1837+
Current vital statistics in Town Reports: No
Compiled Town Vital Records: Yes
Dates: 1887-1935 [Births only] Call No. G 929.3742 N17n
City Directories: Yes
Call No. Date
917.428 N17 [1850, 1853]
917.428 N17g [1864, 1866, 1868-1870, 1872-1885, 1887-1889, 1891-1984]
917.428 N17ga [1988-1991] Cemetery Records: No
Town Register: No
Other resources can be found in the left-hand sidebar. See also their page about special collections.
Other Libraries
Historical Societies
Other Archives
For holdings in other archives, try ArchiveGrid and NUCMC.
Newspapers
To find historical newspapers, which may have articles about the poor farms and the people there, use resources like these:
Genealogical Societies
Try contacting local people who have research in the area. See the FamilySearch Wiki article New Hampshire Societies, try local pages on Facebook, etc.
Publications
Search for articles about New Hampshire's poor farms in genealogical publications and in academic works, and look at the bibliograpies to see what sources the researchers were able to find. You can locate articles by using:
In addition to these, try looking for research guides, finding aids, books about the poor farms, and local histories using:
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