Also I have tried to add a GIN index for better text search as below,
CREATE INDEX idx_fnmae_lname_gin_composite ON wldbowner.member USING gin (lower((((fname)::text || ' '::text) || (lname)::text)) rdsadmin.gin_trgm_ops);
You index does not match your query:
((lower(unaccent_string((lname)::text)) ~~ 'info%'::text) AND (lower(unaccent_string((fname)::text)) ~~ 'travel%'::text))
Your index is not passing the columns through unaccent_string, and it is concatenating the columns while the query is treating them separately. You need to make the index (or indexes, as you might want one for each column) match the query.
If the wildcard is always at the end of the search-pattern strings, you could instead use btree indexes with text_pattern_ops.