You can create a migration for adding a tsv field to your table, as in the following example:

defmodule Your.Repo.Migrations.AddTsVectorsToUsers do
  use Ecto.Migration

  def change do
    alter table("users") do
      add :full_name_tsv,      :tsvector
    end

    execute "UPDATE users SET full_name_tsv = to_tsvector('english', COALESCE(full_name, ''))"
    create index(:users, [:full_name_tsv], using: :gin)

    execute """
    CREATE TRIGGER users_name_tsv_trigger BEFORE INSERT OR UPDATE
    ON users FOR EACH ROW EXECUTE PROCEDURE
    tsvector_update_trigger(full_name_tsv, 'pg_catalog.english', full_name);
    """
  end
end

This will create a tsv field and populate it from the existing field full_name and also index it. It then creates a trigger so that whenever you insert or update an existing record, it recomputes the ts vector.

Then you can define a macro, e.g.:

defmacro tsquery(field, text) do
    quote do
      fragment("?::tsvector @@ to_tsquery('english', ?)", unquote(field), unquote(text))
    end
end

I also have a helper function to split text input into a tsvector sequence, which also replaces invalid characters (only for my use case you might need a different set)

@spec split_names_for_tsquery(String.t()) :: String.t()
def split_names_for_tsquery(text) do
    String.split(text, " ", trim: true)
    |> Enum.reject(fn(text) -> Regex.match?(~r/\(|\)\[|\]\{|\}/, text) end)
    |> Enum.map(fn(token) ->  token <> ":*" end)
    |> Enum.intersperse(" & ")
    |> Enum.join
end

And I use it, similarly to:

import HelperModule, only: [tsquery: 2, split_names_for_tsquery: 1]
#....
n_text = split_names_for_tsquery(text)

Users
|> where([a], tsquery(a.full_name_tsv, ^n_text))
|> order_by([a], asc: a.full_name)
|> limit(^limit_results)

AFAIK the vulnerability lies not in an attacker being able to craft arbitrary SQL, since the input value is being quoted by Ecto, but some LIKE patterns can be quite heavy on the database and cause DoS if used maliciously.

Have a look here: LIKE injection - The GitHub Blog

In short, you might want to quote or strip the wildcard characters before passing them to Ecto.

LIKE and the Postgres fulltext search have some important differences, you should carefully consider what you actually need. The fulltext search processes the text and e.g. by default removes the ending of words (stemming) and also removes certain special characters.

You can speed up LIKE queries with a trigram index, speed is not necessarily a reason to use the fulltext search. It really depends on what kind of searches you want to allow.