Like any question on the legality of an issue, the answer is "it depends."

Under the Revised Code of Washington, Chapter 7.48.305

Notwithstanding any other provision of this chapter, agricultural activities conducted on farmland and forest practices, if consistent with good agricultural and forest practices and established prior to surrounding nonagricultural and nonforestry activities, are presumed to be reasonable and shall not be found to constitute a nuisance unless the activity or practice has a substantial adverse effect on public health and safety. RCW 7.48.305

and,

For the purposes of RCW 7.48.305 only:
(1) "Agricultural activity" means a condition or activity which occurs on a farm in connection with the commercial production of farm products and includes, but is not limited to, marketed produce at roadside stands or farm markets; noise; odors; dust; fumes; operation of machinery and irrigation pumps; movement, including, but not limited to, use of current county road ditches, streams, rivers, canals, and drains, and use of water for agricultural activities; ground and aerial application of seed, fertilizers, conditioners, and plant protection products; keeping of bees for production of agricultural or apicultural products; employment and use of labor; roadway movement of equipment and livestock; protection from damage by wildlife; prevention of trespass; construction and maintenance of buildings, fences, roads, bridges, ponds, drains, waterways, and similar features and maintenance of streambanks and watercourses; and conversion from one agricultural activity to another, including a change in the type of plant-related farm product being produced. The term includes use of new practices and equipment consistent with technological development within the agricultural industry.
(2) "Farm" means the land, buildings, freshwater ponds, freshwater culturing and growing facilities, and machinery used in the commercial production of farm products.
(3) "Farmland" means land or freshwater ponds devoted primarily to the production, for commercial purposes, of livestock, freshwater aquacultural, or other farm products.
(4) "Farm product" means those plants and animals useful to humans and includes, but is not limited to, forages and sod crops, dairy and dairy products, poultry and poultry products, livestock, including breeding, grazing, and recreational equine use, fruits, vegetables, flowers, seeds, grasses, trees, freshwater fish and fish products, apiaries and apiary products, equine and other similar products, or any other product which incorporates the use of food, feed, fiber, or fur. RCW 7.48.310

If you are keeping your bees according the State Laws and local codes: City of Spokane, City of Spokane Valley - Supplemental Use Regulations for Residential Zones, Spokane County Zoning Code (7.07 MB), then beekeeping is not - on its face - a nuisance unless the activity or practice has a substantial adverse effect on public health and safety.