Sure, if they qualify under the rules of their state. Federal rules disallow Medicaid to new immigrants for 5 years, but many states use state funds to provide it to people even in the first 5 years.
If they immigrated in a family-based category, one or more sponsors filled out Affidavits of Support for them, and while the obligations still apply (until the immigrant becomes a citizen, earns 40 quarters of Social Security credits, dies, etc.), if the state "deems" the sponsor's income and assets to be available to the immigrant, then theoretically the sponsor(s) could be liable for the costs incurred by the immigrant's use of need-based benefits like Medicaid. In practice, pretty much no states pursue this.
