The following query:

PREFIX qb: <http://purl.org/linked-data/cube#>
SELECT DISTINCT ?subject
WHERE 
{
  ?x qb:component ?subject .
}

deployed via this publicly available endpoint: http://statistics.gov.scot/sparql-beta produces a list of available subjects:

http://statistics.gov.scot/def/component-specification/pupil-attainment/refArea
http://statistics.gov.scot/def/component-specification/pupil-attainment/refPeriod
http://statistics.gov.scot/def/component-specification/pupil-attainment/measureType
http://statistics.gov.scot/def/component-specification/pupil-attainment/count
http://statistics.gov.scot/def/component-specification/pupil-attainment/pupils
http://statistics.gov.scot/def/component-specification/pupil-attainment/ratio

Problem

I would like to filter the available results using regex, similary to the linked example:

PREFIX vcard: <http://www.w3.org/2001/vcard-rdf/3.0#>

SELECT ?g
WHERE
{ ?y vcard:Given ?g .
  FILTER regex(?g, "r", "i") }

when attempted:

PREFIX qb: <http://purl.org/linked-data/cube#>
SELECT DISTINCT ?subject
WHERE 
{
  ?a qb:component ?subject .
  FILTER regex(?subject, "p", "i")
}

the query returns table with no rows. Is there a way I could filter the obtained values?

I think that you should call the str function on ?subject, it is not a string but rather a URI, so first you have to convert it to a string.

This should work:

PREFIX qb: <http://purl.org/linked-data/cube#>
SELECT DISTINCT ?subject
WHERE 
{
  ?a qb:component ?subject .
  FILTER regex(str(?subject), "p", "i")
}