first_nameカラム、last_nameカラムからフルネーム検索するメソッド

Controller:

class ProfilesController < ApplicationController

  def search
    @profiles = Profile.search_by_full_name(search_params[:name])
  end

  private

  def search_params
    params.require(:search_profile).permit(:name)
  end
end

Model:

class Profile < ApplicationRecord

  def self.search_by_full_name(query)
    query.delete!(' ')
    where("CONCAT_WS('', first_name, last_name) LIKE ?", "%#{query}%")
  end
end

いろいろ応用できそう